rethinkingINLA_HW2
Ania Kawiecki
8/4/2020
Statistical rethinking homework solutions
library(tidyverse)
library(rethinking)
library(dagitty)
library(INLA)
library(knitr)
library(stringr)Intro to linear prediction from Statistical Rethinking 2nd edition Chapter 4.
Finding the posterior distribution
Bayesian updating will allow us to consider every possible combination of values for μ and σ and to score each combination by its relative plausibility, in light of the data. These relative plausibilities are the posterior probabilities of each combination of values μ, σ. Posterior plausibility provides a measure of the logical compatibility of each possible distribution with the data and model.
The thing to worry about is keeping in mind that the “estimate” here will be the entire posterior distribution, not any point within it. And as a result, the posterior distribution will be a distribution of Gaussian distributions. Yes, a distribution of distributions.
The prior for μ is a broad Gaussian prior, centered on 178cm, with 95% of probability between 178 ± 40.
The weights that interest us are all adult weights, so we can analyze only the adults and make an okay linear approximation.
data(Howell1)
d <- Howell1
d2 <- d[ d$age >= 18 , ]
xbar <- mean(d2$weight) The golem is assuming that the average height (not each individual height) is almost certainly between 140 cm and 220 cm
#plot mean prior
curve( dnorm( x , 178 , 20 ) , from=100 , to=250 )
The σ prior is a truly flat prior, a uniform one, that functions just to constrain σ to have positive probability between zero and 50cm. A standard deviation of 50cm would imply that 95% of individual heights lie within 100cm of the average height. That’s a very large range.
#plot sd prior
curve( dunif( x , 0 , 50 ) , from=-10 , to=60 )
The prior predictive simulation is an essential part of your modeling. Once you’ve chosen priors for h, μ, and σ, these imply a joint prior distribution of individual heights. By simulating from this distribution, you can see what your choices imply about observable height. This helps you diagnose bad choices. Lots of conventional choices are indeed bad ones, and we’ll be able to see this through prior predictive simulations.
Okay, so how to do this? You can quickly simulate heights by sampling from the prior.
sample_mu <- rnorm( 1e4 , 178 , 20 )
sample_sigma <- runif( 1e4 , 0 , 50 )
prior_h <- rnorm( 1e4 , sample_mu , sample_sigma )
#prior_h
dens( prior_h )
This is the expected distribution of heights, averaged over the prior. Notice that the prior probability distribution of height is not itself Gaussian. This is okay. The distribution you see is not an empirical expectation, but rather the distribution of relative plausibilities of different heights, before seeing the data.
μ i = α + β ( x i − ̄x )
What this tells the regression golem is that you are asking two questions about the mean of the outcome.
What is the expected height when xi = ̄x? The parameter α answers this question, because when xi = ̄x, μi = α. For this reason, α is often called the intercept. But we should think not in terms of some abstract line, but rather in terms of the meaning with respect to the observable variables.
What is the change in expected height, when xi changes by 1 unit? The parameter β answers this question. It is often called a “slope,” again because of the abstract line. Better to think of it as a rate of change in expectation. Jointly these two parameters ask the golem to find a line that relates x to h, a line that passes through α when xi = ̄x and has slope β. That is a task that golems are very good at. It’s up to you, though, to be sure it’s a good question.
The goal is to simulate heights from the model, using only the priors. First, let’s consider a range of weight values to simulate over. The range of observed weights will do fine. Then we need to simulate a bunch of lines, the lines implied by the priors for α and β. Now we have 100 pairs of α and β values. Now to plot the lines:
set.seed(2971)
N <- 100 # 100 lines
a <- rnorm( N , 178 , 20 )
b <- rnorm( N , 0 , 10 )
plot( NULL , xlim=range(d2$weight) , ylim=c(-100,400) ,
xlab="weight" , ylab="height" )
abline( h=0 , lty=2 )
abline( h=272 , lty=1 , lwd=0.5 )
mtext( "b ~ dnorm(0,10)" )
xbar <- mean(d2$weight)
for ( i in 1:N ) curve( a[i] + b[i]*(x - xbar) ,
from=min(d2$weight) , to=max(d2$weight) , add=TRUE ,
col=col.alpha("black",0.2) )
If the logarithm of β is normal, then β itself is strictly positive. The reason is that exp(x) is greater than zero for any real number x. This is the reason that Log-Normal priors are commonplace. They are an easy way to enforce positive relationships
set.seed(2971)
N <- 100 # 100 lines
a <- rnorm( N , 178 , 20 )
b <- rlnorm( N , 0 , 1 )
plot( NULL , xlim=range(d2$weight) , ylim=c(-100,400) ,
xlab="weight" , ylab="height" )
abline( h=0 , lty=2 )
abline( h=272 , lty=1 , lwd=0.5 )
mtext( "b ~ dnorm(0,10)" )
xbar <- mean(d2$weight)
for ( i in 1:N ) curve( a[i] + b[i]*(x - xbar) ,
from=min(d2$weight) , to=max(d2$weight) , add=TRUE ,
col=col.alpha("black",0.2) )
Overthinking: Logs and exps, oh my. My experience is that many natural and social scientists have naturally forgotten whatever they once knew about logarithms. Logarithms appear all the time in applied statistics. You can usefully think of y = log(x) as assigning to y the order of magnitude of x. The function x = exp(y) is the reverse, turning a magnitude into a value. These definitions will make a mathematician shriek. But much of our computational work relies only on these intuitions. These definitions allow the Log-Normal prior for β to be coded another way. Instead of defining a parameter β, we define a parameter that is the logarithm of β and then assign it a normal distribution. Then we can reverse the logarithm inside the linear model. It looks like this:
m4.3b <- quap( alist(
height ~ dnorm( mu , sigma ) ,
mu <- a + exp(log_b)*( weight - xbar ), a ~ dnorm( 178 , 20 ) ,
log_b ~ dnorm( 0 , 1 ) ,
sigma ~ dunif( 0 , 50 )
) , data=d2 )Note the exp(log_b) in the definition of mu. This is the same model as m4.3. It will make the same predictions. But instead of β in the posterior distribution, you get log(β). It is easy to translate between the two, because β = exp(log(β)). In code form: b <- exp(log_b).
Interpreting the posterior distribution. One trouble with statistical models is that they are hard to understand. Once you’ve fit the model, it can only report posterior distribution. This is the right answer to the question you asked. But it’s your responsibility to process the answer and make sense of it.
m4.3 <- quap(
alist(
height ~ dnorm( mu , sigma ) ,
mu <- a + b*( weight - xbar ) ,
a ~ dnorm( 178 , 20 ) ,
b ~ dlnorm( 0 , 1 ) ,
sigma ~ dunif( 0 , 50 )
), data=d2 )
precis(m4.3)## mean sd 5.5% 94.5%
## a 154.6013713 0.2703075 154.1693677 155.0333748
## b 0.9032763 0.0419236 0.8362742 0.9702783
## sigma 5.0718775 0.1911545 4.7663757 5.3773792The first row gives the quadratic approximation for α, the second the approximation for β, and the third approximation for σ. Let’s try to make some sense of them.
Let’s focus on b (β), because it’s the new parameter. Since β is a slope, the value 0.90 can be read as a person 1 kg heavier is expected to be 0.90 cm taller. 89% of the posterior probability lies between 0.84 and 0.97. That suggests that β values close to zero or greatly above one are highly incompatible with these data and this model. It is most certainly not evidence that the relationship between weight and height is linear, because the model only considered lines. It just says that, if you are committed to a line, then lines with a slope around 0.9 are plausible ones.
You can see the covariances among the parameters with vcov:
round( vcov( m4.3 ) , 3 )## a b sigma
## a 0.073 0.000 0.000
## b 0.000 0.002 0.000
## sigma 0.000 0.000 0.037# shows both the marginal posteriors and the covariance.
pairs(m4.3)
Very little covariation among the parameters in this case. The lack of covariance among the parameters results from centering.
Plotting posterior inference against the data. It’s almost always much more useful to plot the posterior inference against the data. Not only does plotting help in interpreting the posterior, but it also provides an informal check on model assumptions. When the model’s predictions don’t come close to key observations or patterns in the plotted data, then you might suspect the model either did not fit correctly or is rather badly specified. But even if you only treat plots as a way to help in interpreting the posterior, they are invaluable.
Each point in this plot is a single individual. The black line is defined by the mean slope β and mean intercept α = the posterior mean line. It looks highly plausible. But there an infinite number of other highly plausible lines near it. Let’s draw those too.
plot( height ~ weight , data=d2 , col=rangi2 )
post <- extract.samples( m4.3 )
a_map <- mean(post$a)
b_map <- mean(post$b)
curve( a_map + b_map*(x - xbar) , add=TRUE )
post <- extract.samples( m4.3 ) = Each row is a correlated random sample from the joint posterior of all three parameters, using the covariances provided by vcov(m4.3). The paired values of a and b on each row define a line. The average of very many of these lines is the posterior mean line. But the scatter around that average is meaningful, because it alters our confidence in the relationship between the predictor and the outcome.
Let’s display a bunch of these lines, so you can see the scatter. This lesson will be easier to appreciate, if we use only some of the data to begin. Then you can see how adding in more data changes the scatter of the lines. So we’ll begin with just the first 10 cases in d2. The following code extracts the first 10 cases and re-estimates the model:
N <- 10
dN <- d2[ 1:N , ]
mN <- quap(
alist(
height ~ dnorm( mu , sigma ) ,
mu <- a + b*( weight - mean(weight) ) ,
a ~ dnorm( 178 , 20 ) ,
b ~ dlnorm( 0 , 1 ) ,
sigma ~ dunif( 0 , 50 )
) , data=dN )Now let’s plot 20 of these lines, to see what the uncertainty looks like.
# extract 20 samples from the posterior
post <- extract.samples( mN , n=20 )
# display raw data and sample size
plot( dN$weight , dN$height ,
xlim=range(d2$weight) , ylim=range(d2$height) ,
col=rangi2 , xlab="weight" , ylab="height" )
mtext(concat("N = ",N))
# plot the lines, with transparency
for ( i in 1:20 )
curve( post$a[i] + post$b[i]*(x-mean(dN$weight)) ,
col=col.alpha("black",0.3) , add=TRUE )
Increae the amounts of data. Notice that the cloud of regression lines grows more compact as the sample size increases. This is a result of the model growing more confident about the location of the mean.
N <- 352
dN <- d2[ 1:N , ]
mN <- quap(
alist(
height ~ dnorm( mu , sigma ) ,
mu <- a + b*( weight - mean(weight) ) ,
a ~ dnorm( 178 , 20 ) ,
b ~ dlnorm( 0 , 1 ) ,
sigma ~ dunif( 0 , 50 )
) , data=dN )
# extract 20 samples from the posterior
post <- extract.samples( mN , n=20 )
# display raw data and sample size
plot( dN$weight , dN$height ,
xlim=range(d2$weight) , ylim=range(d2$height) ,
col=rangi2 , xlab="weight" , ylab="height" )
mtext(concat("N = ",N))
# plot the lines, with transparency
for ( i in 1:20 )
curve( post$a[i] + post$b[i]*(x-mean(dN$weight)) ,
col=col.alpha("black",0.3) , add=TRUE )
Focus for the moment on a single weight value, say 50 kilograms. You can quickly make a list of 10,000 values of μ (height)for an individual who weighs 50 kilograms, by using your samples from the posterior.
μ i = α + β ( x i − ̄x ) The value of xi in this case is 50.
mu_at_50 is a vector of predicted means, one for each random sample from the posterior. Since joint a and b went into computing each, the variation across those means incorporates the uncertainty in and correlation between both parameters. It might be helpful at this point to actually plot the density for this vector of means.
Since the components of μ have distributions, so too does μ. And since the distributions of α and β are Gaussian, so to is the distribution of μ (adding Gaussian distributions always produces a Gaussian distribution). Since the posterior for μ is a distribution, you can find intervals for it, just like for any posterior distribution. The central 89% of the ways for the model to produce the data place the average height between about 159 cm and 160 cm (conditional on the model and data), assuming the weight is 50 kg.
post <- extract.samples( m4.3 )
mu_at_50 <- post$a + post$b * ( 50 - xbar )
dens( mu_at_50 , col=rangi2 , lwd=2 , xlab="mu|weight=50" )
PI( mu_at_50 , prob=0.89 )## 5% 94%
## 158.5677 159.6695That’s good so far, but we need to repeat the above calculation for every weight value on the horizontal axis, not just when it is 50 kg. We want to draw 89% intervals around the average slope.
This is made simple by strategic use of the link function, a part of the rethinking package. What link will do is take your quap approximation, sample from the posterior distribution, and then compute μ for each case in the data and sample from the posterior distribution. Here’s what it looks like for the data you used to fit the model:
mu <- link( m4.3 )
str(mu)## num [1:1000, 1:352] 157 157 157 157 157 ...You end up with a big matrix of values of μ. Each row is a sample from the posterior distribu- tion. There are 352 rows in d2, corresponding to 352 individuals. So there are 352 columns in the matrix mu above. link takes 1000 samples of the posterior dist. for every value in the data (of weight in this case).
this is what the rethinking::link function does:
post <- extract.samples(m4.3)
mu.link <- function(weight) post$a + post$b*( weight - xbar )
weight.seq <- seq( from=25 , to=70 , by=1 )
mu <- sapply( weight.seq , mu.link )
mu.mean <- apply( mu , 2 , mean )
mu.CI <- apply( mu , 2 , PI , prob=0.89 )The function link provides a posterior distribution of μ for each case we feed it. So above we have a distribution of μ for each individual in the original data. We actually want something slightly different: a distribution of μ for each unique weight value on the horizontal axis.
# define sequence of weights to compute predictions for
# these values will be on the horizontal axis
weight.seq <- seq( from=25 , to=70 , by=1 )
# use link to compute mu
# for each sample from posterior
# and for each weight in weight.seq
mu <- link( m4.3 , data=data.frame(weight=weight.seq) )
str(mu)## num [1:1000, 1:46] 137 136 138 136 136 ...And now there are only 46 columns in mu, because we fed it 46 different values for weight.
To visualize what you’ve got here, let’s plot the distribution of μ values at each height.
# use type="n" to hide raw data
plot( height ~ weight , d2 ) #, type="n" )
# loop over samples and plot each mu value
for ( i in 1:100 )
points( weight.seq , mu[i,] , pch=16 , col=col.alpha(rangi2,0.1) )
At each weight value in weight.seq, a pile of computed μ values are shown. Each of these piles is a Gaussian distribution, the amount of uncertainty in μ depends upon the value of weight.
The final step is to summarize the distribution for each weight value. We’ll use apply, which applies a function of your choice to a matrix.
mu.mean contains the average μ at each weight value, and mu.PI contains 89% lower and upper bounds for each weight value.
# summarize the distribution of mu
#compute the mean of each column (dimension “2”) of the matrix mu.
mu.mean <- apply( mu , 2 , mean )
mu.PI <- apply( mu , 2 , PI , prob=0.89 )
# plot raw data
# fading out points to make line and interval more visible
plot( height ~ weight , data=d2 , col=col.alpha(rangi2,0.5) )
# plot the MAP line, aka the mean mu for each weight
lines( weight.seq , mu.mean )
# plot a shaded region for 89% PI
shade( mu.PI , weight.seq )
To summarize, here’s the recipe for generating predictions and intervals from the poste- rior of a fit model.
Use link to generate distributions of posterior values for μ. The default behavior of link is to use the original data, so you have to pass it a list of new horizontal axis values you want to plot posterior predictions across.
Use summary functions like mean or PI to find averages and lower and upper bounds of μ for each value of the predictor variable.
Finally,use plotting functions like lines and shade to draw the lines and intervals. Or you might plot the distributions of the predictions, or do further numerical calculations with them. It’s really up to you.
This recipe works for every model we fit in the book. As long as you know the structure of the model—how parameters relate to the data—you can use samples from the posterior to describe any aspect of the model’s behavior.
Prediction intervals Now let’s walk through generating an 89% prediction in- terval for actual heights, not just the average height, μ. This means we’ll incorporate the standard deviation σ and its uncertainty as well. Remember, the first line of the statistical model here is:
hi ∼ Normal(μi, σ)
What you’ve done so far is just use samples from the posterior to visualize the uncertainty in μi, the linear model of the mean. But actual predictions of heights depend also upon the distribution in the first line. The Gaussian distribution on the first line tells us that the model expects observed heights to be distributed around μ, not right on top of it. And the spread around μ is governed by σ. All of this suggests we need to incorporate σ in the predictions somehow.
Here’s how you do it. Imagine simulating heights. For any unique weight value, you sample from a Gaussian distribution with the correct mean μ for that weight, using the correct value of σ sampled from the same posterior distribution. If you do this for every sample from the posterior, for every weight value of interest, you end up with a collection of simulated heights that embody the uncertainty in the posterior as well as the uncertainty in the Gaussian distribution of heights. There is a tool called sim which does this.
This matrix is much like the earlier one, mu, but it contains simulated heights, not distributions of plausible average height, μ.
We can summarize these simulated heights in the same way we summarized the distributions of μ, by using the PI function over apply.
sim.height <- sim( m4.3 , data=list(weight=weight.seq) )
#This matrix is much like the earlier one, mu, but it contains simulated heights, not distributions of plausible average height, μ.
str(sim.height)## num [1:1000, 1:46] 145 126 137 137 142 ...#height.PI contains the 89% posterior prediction interval of observable (according to the model) heights, across the values of weight in weight.seq
height.PI <- apply( sim.height , 2 , PI , prob=0.89 )Let’s plot everything we’ve built up: (1) the average line (2) the shaded region of 89% plausible μ (3) the boundaries of the simulated heights the model expects.
# plot raw data
plot( height ~ weight , d2 , col=col.alpha(rangi2,0.5) )
# draw MAP line : the average line of all the mean heights expected by the linear model for each weight
#mu.link <- function(weight) post$a + post$b*( weight - xbar )
#weight.seq <- seq( from=25 , to=70 , by=1 )
#mu <- sapply( weight.seq , mu.link )
#mu.mean <- apply( mu , 2 , mean )
lines( weight.seq , mu.mean )
# draw 89% region for line
shade( mu.PI , weight.seq )
# draw PI region for simulated heights:
#the area within which the model expects to find 89% of actual heights in the population, at each weight.
shade( height.PI , weight.seq )
89% prediction interval for height, as a function of weight. The solid line is the average line for the mean height at each weight. The two shaded regions show different 89% plausible regions. The narrow shaded in- terval around the line is the distribution of μ. The wider shaded region represents the region within which the model expects to find 89% of actual heights in the population, at each weight.
this is what the rethinking::sim function does
#extract samples automatically extracts 1000 from posterior dist
post <- extract.samples(m4.3)
weight.seq <- 25:70
sim.function <- function(weight)
rnorm(
n=nrow(post) ,
mean=post$a + post$b*( weight - xbar ) ,
sd=post$sigma )
sim.height <- sapply( weight.seq , sim.function )
##compute the mean of each column (dimension “2”) of the matrix mu.
height.interval <- apply(sim.height, 2, quantile, c( 0.05 , 0.94 ))
#example of compatibility interval for 1st col (first weight)
#quantile(sim.height[,1],c( 0.05 , 0.94 ) )
# plot raw data
plot( height ~ weight , d2 , col=col.alpha(rangi2,0.5) )
# draw MAP line
lines( weight.seq , mu.mean )
# draw HPDI region for line
shade( mu.PI , weight.seq )
# draw PI region for simulated heights
shade( height.interval, weight.seq )
Rethinking: Two kinds of uncertainty. In the procedure above, we encountered both uncertainty in parameter values and uncertainty in a sampling process. These are distinct concepts, even though they are processed much the same way and end up blended together in the posterior predictive simu- lation. The posterior distribution is a ranking of the relative plausibilities of every possible combina- tion of parameter values. The distribution of simulated outcomes, like height, is instead a distribution that includes sampling variation from some process that generates Gaussian random variables. This sampling variation is still a model assumption. It’s no more or less objective than the posterior distribution. Both kinds of uncertainty matter, at least sometimes. But it’s important to keep them straight, because they depend upon different model assumptions.
HOMEWORK 2
1.
The weights listed below were recorded in the !Kung census, but heights were not recorded for these individuals. Provide predicted heights and 89% compatibility intervals for each of these individuals, using model-based predictions.
ind <- 1:5
weight <- c(45, 40, 65, 31, 53)
expected_height <- NA
interval <- NA
kung <- bind_cols(ind, weight, expected_height, interval)## New names:
## * NA -> ...1
## * NA -> ...2
## * NA -> ...3
## * NA -> ...4colnames(kung) <- c("individual", "weight", "expected_height", "89%interval")
kable(kung)| individual | weight | expected_height | 89%interval |
|---|---|---|---|
| 1 | 45 | NA | NA |
| 2 | 40 | NA | NA |
| 3 | 65 | NA | NA |
| 4 | 31 | NA | NA |
| 5 | 53 | NA | NA |
1. rethinking
m4.3 <- quap(
alist(
height ~ dnorm( mu , sigma ) ,
mu <- a + b*( weight - xbar ) ,
a ~ dnorm( 178 , 20 ) ,
b ~ dlnorm( 0 , 1 ) ,
sigma ~ dunif( 0 , 50 )
), data=d2 )
precis(m4.3)## mean sd 5.5% 94.5%
## a 154.6013672 0.27030763 154.1693634 155.0333710
## b 0.9032809 0.04192363 0.8362788 0.9702829
## sigma 5.0718803 0.19115473 4.7663781 5.3773825Now we need posterior predictions for each case in the table. Easiest way to do this is to use sim. We need sim, not just link, because we are trying to predict an individual’s height. So the relevant compatibility interval includes the Gaussian variance from sigma. If you provided only the compatibility interval for μ, that’s okay. But be sure you understand the difference.
Solution using rethinking functions
dat <- data.frame(weight= c(45, 40, 65, 31, 53))
h_sim <- sim( m4.3 , data=dat )
Eh <- apply(h_sim,2,mean)
h_ci <- apply(h_sim,2,PI,prob=0.89)
dat$Eh <- Eh
dat$L89 <- h_ci[1,]
dat$U89 <- h_ci[2,]
round(dat,1)## weight Eh L89 U89
## 1 45 154.6 146.6 163.0
## 2 40 149.9 141.8 157.9
## 3 65 172.7 164.5 181.1
## 4 31 141.9 133.6 150.2
## 5 53 162.1 154.2 169.8Solution using base r functions
weight= c(45, 40, 65, 31, 53)
sim.hw2.fun <- function(weight, model) {
post.hw2 = extract.samples(model)
rnorm(
n= nrow(post),
mean= post$a + post$b*(weight - xbar),
sd= post$sigma)
}
sim.hw2 <- sapply(weight, sim.hw2.fun , m4.3)
hw2.mean <- apply(sim.hw2,2, mean)
hw2.ci <- apply(sim.hw2,2, quantile, c(0.05, 0.95))
hw2.1 <- data.frame(weight= c(45, 40, 65, 31, 53)) %>%
mutate(expected_height= hw2.mean,
LCI= hw2.ci[1],
UCI = hw2.ci[2])
print(hw2.1)## weight expected_height LCI UCI
## 1 45 154.5601 146.0763 162.9388
## 2 40 150.1655 146.0763 162.9388
## 3 65 172.7602 146.0763 162.9388
## 4 31 141.9387 146.0763 162.9388
## 5 53 161.8957 146.0763 162.93881. inla
https://www.flutterbys.com.au/stats/tut/tut12.10.html
https://people.bath.ac.uk/jjf23/brinla/hubble.html#inla-weakly-or-non-informative
https://people.bath.ac.uk/jjf23/brinla/chicago.html
https://haakonbakka.bitbucket.io/btopic112.html
library(devtools)## Loading required package: usethisinstall_github("julianfaraway/brinla")## Skipping install of 'brinla' from a github remote, the SHA1 (95181536) has not changed since last install.
## Use `force = TRUE` to force installationlibrary(brinla)
library("inlabru")The default mean and precision for fixed effects are:
inla.set.control.fixed.default()[c('mean','prec')]## $mean
## [1] 0
##
## $prec
## [1] 0.001We see that the default prior on beta is normal with mean zero and precision 0.001. The precision is the inverse of the variance. We convert this to SD:
sqrt(1/0.001)## [1] 31.62278We wish to predict the response at a new set of inputs. We add a case for the new inputs (weight= c(45, 40, 65, 31, 53)) and set the response to missing (height=NA):
data(Howell1)
d <- Howell1
d2 <- d[ d$age >= 18 , ]
dat <- data.frame(weight= c(45, 40, 65, 31, 53))
xbar <- mean(d2$weight)
new_w <- bind_cols(weight, NA)## New names:
## * NA -> ...1
## * NA -> ...2colnames(new_w) <- c("weight", "height")
# add the weight values of interest to the dataframe
d1.i <- d2 %>%
select(c("weight", "height")) %>%
rbind(new_w) %>%
mutate(w= weight-xbar) %>%
select( c("height", "w"))
#indices of the weights with missing values of height
d1.i.na <- which(is.na(d1.i$height))
#We need to set the control.predictor to compute the posterior means of the linear predictors
m1.i<- inla(height ~ w, data= d1.i,
control.fixed = list(
mean= 0,
prec= 1,
mean.intercept= 178,
prec.intercept= 1/(20^2)),# sd = 20 --> precision =1/variance --> 1/(sd^2)
control.compute = list(config= TRUE),
control.predictor=list(compute=TRUE)
)
# posterior means and SDs from the INLA fit
m1.i$summary.fixed[,1:2]## mean sd
## (Intercept) 154.6013921 0.27107172
## w 0.9034319 0.04200942# the summary statistics of the fitted values for the weight values of interest can be shown using index of the values with the missing height
m1.i$summary.fitted.values[d1.i.na, ]## mean sd 0.025quant 0.5quant 0.975quant mode
## fitted.Predictor.353 154.6100 0.2710729 154.0782 154.6100 155.1417 154.6100
## fitted.Predictor.354 150.0928 0.3426798 149.4206 150.0928 150.7651 150.0927
## fitted.Predictor.355 172.6786 0.8831834 170.9456 172.6787 174.4108 172.6789
## fitted.Predictor.356 141.9619 0.6472126 140.6924 141.9619 143.2316 141.9617
## fitted.Predictor.357 161.8374 0.4320724 160.9897 161.8375 162.6849 161.83752.
Model the relationship between height(cm) and the natural logarithm of weight (log-kg): log(weight). Use the entire Howell1 data frame, all 544 rows, adults and non-adults. Use any model type from Chapter 4 that you think useful: an ordinary linear regression, a polynomial or a spline. Plot the posterior predictions against the raw data.
2.rethinking
data(Howell1)
d <- Howell1
d$log_weight <- log(d$weight)
xbar <- mean(d$log_weight)
m2 <- quap(
alist(
height ~ dnorm( mu , sigma ) ,
mu <- a + b*( log_weight - xbar ) ,
a ~ dnorm( 178 , 20 ) ,
b ~ dlnorm( 0 , 1 ) ,
sigma ~ dunif( 0 , 50 )
), data=d )
precis(m2)## mean sd 5.5% 94.5%
## a 138.268410 0.2201360 137.916591 138.620230
## b 47.071138 0.3826317 46.459618 47.682657
## sigma 5.134716 0.1556694 4.885926 5.383506plot( d$weight , d$height , col=col.alpha(rangi2,0.7) )
x_seq <- log(1:60)
mu <- sim( m2 , data=list(log_weight=x_seq) )
mu_mean <- apply(mu,2,mean)
mu_ci <- apply(mu,2,PI,0.99)
lines( exp(x_seq) , mu_mean )
shade( mu_ci , exp(x_seq) )
You could certainly do better—the trend is under-predicting in the mid ages. But just taking the log of weight does most of the work. Why? It’ll help to think of a human body as a cylinder. Roughly. The weight of a cylinder is proportional to its volume. And the volume of a cylinder is: V = πr2h where r is the radius and h is the height. As the cylinder, uh human, gets taller, the radius gets bigger. So we can just say the radius is some fraction α of the height: Substituting that in: r = αh V = πα2h3 = kh3 where k = πα2 is just some proportionality constant.
2.INLA
library(brinla)
data(Howell1)
d <- Howell1
log_wplot <- log(1:60)
new_logw <- bind_cols(log_wplot, NA)## New names:
## * NA -> ...1
## * NA -> ...2colnames(new_logw) <- c("log_weight", "height")
# add the weight values of interest to the dataframe
d2.i <- d %>%
mutate(log_weight= log(weight)) %>%
select(c("log_weight", "height")) %>%
rbind(new_logw) %>%
mutate(xbar= mean(log_weight),
log_w= log_weight - xbar)
#We need to set the control.predictor to compute the posterior means of the linear predictors
m2.i<- inla(height ~ log_w, data= d2.i,
control.fixed = list(
mean= 0,
prec= 1,
mean.intercept= 178,
prec.intercept= 1/(20^2)),# sd = 20 --> precision =1/variance --> 1/(sd^2)
control.compute = list(config= TRUE),
control.predictor=list(compute=TRUE)
)
# posterior means and SDs from the INLA fit
m2.i$summary.fixed[,1:2]## mean sd
## (Intercept) 137.8410 0.7915664
## log_w 16.3662 1.3418096#indices of the weights with missing values of height
d2.i.na <- which(is.na(d2.i$height))
# the summary statistics of the fitted values for the weight values of interest can be shown using index of the values with the missing height
m2.i$summary.fitted.values[d2.i.na, ]## mean sd 0.025quant 0.5quant 0.975quant
## fitted.Predictor.545 81.98959 4.6919164 72.92648 81.99131 91.16749
## fitted.Predictor.546 93.33351 3.7781530 86.03917 93.33182 100.73021
## fitted.Predictor.547 99.96928 3.2479897 93.70162 99.96524 106.33486
## fitted.Predictor.548 104.67743 2.8750126 99.13237 104.67149 110.31897
## fitted.Predictor.549 108.32935 2.5883628 103.33989 108.32176 113.41437
## fitted.Predictor.550 111.31319 2.3565293 106.77329 111.30413 115.94764
## fitted.Predictor.551 113.83599 2.1627307 109.67221 113.82559 118.09454
## fitted.Predictor.552 116.02134 1.9969785 112.17964 116.00971 119.95964
## fitted.Predictor.553 117.94896 1.8528574 114.38757 117.93621 121.60854
## fitted.Predictor.554 119.67328 1.7260124 116.35833 119.65950 123.08662
## fitted.Predictor.555 121.23311 1.6133539 118.13713 121.21841 124.42929
## fitted.Predictor.556 122.65713 1.5126262 119.75634 122.64160 125.65898
## fitted.Predictor.557 123.96710 1.4221401 121.24154 123.95087 126.79369
## fitted.Predictor.558 125.17994 1.3405932 122.61241 125.16316 127.84862
## fitted.Predictor.559 126.30908 1.2669768 123.88420 126.29188 128.83583
## fitted.Predictor.560 127.36531 1.2004953 125.06907 127.34786 129.76306
## fitted.Predictor.561 128.35750 1.1405103 126.17723 128.33996 130.63807
## fitted.Predictor.562 129.29295 1.0865054 127.21706 129.27553 131.46833
## fitted.Predictor.563 130.17782 1.0380564 128.19454 130.16071 132.25855
## fitted.Predictor.564 131.01729 0.9948077 129.11590 131.00070 133.01253
## fitted.Predictor.565 131.81580 0.9564569 129.98622 131.79993 133.73450
## fitted.Predictor.566 132.57715 0.9227375 130.81048 132.56222 134.42778
## fitted.Predictor.567 133.30465 0.8934081 131.59228 133.29084 135.09521
## fitted.Predictor.568 134.00119 0.8682436 132.33355 133.98868 135.73933
## fitted.Predictor.569 134.66929 0.8470266 133.03897 134.65824 136.36205
## fitted.Predictor.570 135.31118 0.8295415 133.71048 135.30171 136.96498
## fitted.Predictor.571 135.92885 0.8155733 134.35080 135.92105 137.55145
## fitted.Predictor.572 136.52405 0.8049018 134.96144 136.51799 138.12053
## fitted.Predictor.573 137.09835 0.7973052 135.54599 137.09407 138.67545
## fitted.Predictor.574 137.65319 0.7925575 136.10473 137.65068 139.21538
## fitted.Predictor.575 138.18983 0.7904329 136.64044 138.18908 139.74349
## fitted.Predictor.576 138.70943 0.7907078 137.15528 138.71039 140.25852
## fitted.Predictor.577 139.21305 0.7931610 137.64879 139.21564 140.76235
## fitted.Predictor.578 139.70163 0.7975798 138.12503 139.70577 141.25527
## fitted.Predictor.579 140.17604 0.8037609 138.58260 140.18166 141.73729
## fitted.Predictor.580 140.63710 0.8115108 139.02514 140.64407 142.20994
## fitted.Predictor.581 141.08551 0.8206505 139.45208 141.09375 142.67231
## fitted.Predictor.582 141.52197 0.8310132 139.86489 141.53136 143.12570
## fitted.Predictor.583 141.94710 0.8424462 140.26491 141.95754 143.57045
## fitted.Predictor.584 142.36145 0.8548108 140.65287 142.37285 144.00571
## fitted.Predictor.585 142.76558 0.8679822 141.02928 142.77782 144.43303
## fitted.Predictor.586 143.15997 0.8818469 141.39468 143.17297 144.85264
## fitted.Predictor.587 143.54507 0.8963040 141.74967 143.55875 145.26410
## fitted.Predictor.588 143.92132 0.9112637 142.09510 143.93560 145.66763
## fitted.Predictor.589 144.28912 0.9266464 142.43150 144.30393 146.06342
## fitted.Predictor.590 144.64883 0.9423820 142.75946 144.66409 146.45219
## fitted.Predictor.591 145.00081 0.9584076 143.07934 145.01646 146.83406
## fitted.Predictor.592 145.34537 0.9746681 143.39155 145.36136 147.20939
## fitted.Predictor.593 145.68283 0.9911152 143.69641 145.69910 147.57807
## fitted.Predictor.594 146.01347 1.0077063 143.99390 146.02998 147.94029
## fitted.Predictor.595 146.33757 1.0244043 144.28483 146.35428 148.29626
## fitted.Predictor.596 146.65537 1.0411766 144.56953 146.67224 148.64615
## fitted.Predictor.597 146.96711 1.0579945 144.84816 146.98411 148.99008
## fitted.Predictor.598 147.27303 1.0748330 145.12100 147.29013 149.32822
## fitted.Predictor.599 147.57334 1.0916706 145.38825 147.59051 149.66079
## fitted.Predictor.600 147.86823 1.1084882 145.65020 147.88545 149.98798
## fitted.Predictor.601 148.15791 1.1252692 145.90711 148.17516 150.30992
## fitted.Predictor.602 148.44255 1.1419991 146.15918 148.45981 150.62677
## fitted.Predictor.603 148.72232 1.1586657 146.40663 148.73957 150.93866
## fitted.Predictor.604 148.99739 1.1752580 146.64965 149.01461 151.24574
## mode
## fitted.Predictor.545 82.0982
## fitted.Predictor.546 93.4060
## fitted.Predictor.547 100.0190
## fitted.Predictor.548 104.7101
## fitted.Predictor.549 108.3482
## fitted.Predictor.550 111.3203
## fitted.Predictor.551 113.8329
## fitted.Predictor.552 116.0092
## fitted.Predictor.553 117.9289
## fitted.Predictor.554 119.6462
## fitted.Predictor.555 121.1999
## fitted.Predictor.556 122.6186
## fitted.Predictor.557 123.9240
## fitted.Predictor.558 125.1332
## fitted.Predictor.559 126.2595
## fitted.Predictor.560 127.3138
## fitted.Predictor.561 128.3049
## fitted.Predictor.562 129.2401
## fitted.Predictor.563 130.1255
## fitted.Predictor.564 130.9664
## fitted.Predictor.565 131.7672
## fitted.Predictor.566 132.5314
## fitted.Predictor.567 133.2625
## fitted.Predictor.568 133.9631
## fitted.Predictor.569 134.6358
## fitted.Predictor.570 135.2826
## fitted.Predictor.571 135.9054
## fitted.Predictor.572 136.5058
## fitted.Predictor.573 137.0855
## fitted.Predictor.574 137.6457
## fitted.Predictor.575 138.1876
## fitted.Predictor.576 138.7123
## fitted.Predictor.577 139.2208
## fitted.Predictor.578 139.7141
## fitted.Predictor.579 140.1929
## fitted.Predictor.580 140.6581
## fitted.Predictor.581 141.1103
## fitted.Predictor.582 141.5503
## fitted.Predictor.583 141.9787
## fitted.Predictor.584 142.3960
## fitted.Predictor.585 142.8028
## fitted.Predictor.586 143.1996
## fitted.Predictor.587 143.5868
## fitted.Predictor.588 143.9650
## fitted.Predictor.589 144.3344
## fitted.Predictor.590 144.6956
## fitted.Predictor.591 145.0487
## fitted.Predictor.592 145.3943
## fitted.Predictor.593 145.7327
## fitted.Predictor.594 146.0640
## fitted.Predictor.595 146.3886
## fitted.Predictor.596 146.7069
## fitted.Predictor.597 147.0189
## fitted.Predictor.598 147.3250
## fitted.Predictor.599 147.6254
## fitted.Predictor.600 147.9203
## fitted.Predictor.601 148.2099
## fitted.Predictor.602 148.4943
## fitted.Predictor.603 148.7739
## fitted.Predictor.604 149.0486m2.i.postmean <- bind_cols( new_logw[,1], m2.i$summary.linear.predictor[d2.i.na,]) %>%
select(c("log_weight", "mean", "sd", "0.5quant", "0.975quant"))
names(m2.i.postmean) <- c("log_weight", "mean", "sd", "LCI", "UCI")
m2.i.plot <- ggplot()+
geom_point(data= d, aes(weight, height))+
geom_line(data= m2.i.postmean, aes(exp(log_weight), mean))+
geom_ribbon(data= m2.i.postmean, aes(exp(log_weight), ymin= LCI, ymax= UCI))
m2.i.plot 
3.
Plot the prior predictive distribution for the polynomial regression model in Chapter 4. You can modify the the code that plots the linear regression prior predictive distribution. 20 or 30 parabolas from the prior should suffice to show where the prior probability resides. Can you modify the prior distributions of α, β1, and β2 so that the prior predictions stay within the biologically reasonable outcome space? That is to say: Do not try to fit the data by hand. But do try to keep the curves consistent with what you know about height and weight, before seeing these exact data.